1

The effect of oscillatory flow on nucleation kinetics of butyl paraben

Huaiyu Yang1, Xi Yu2, Vishal Raval1, Yassir Makkawi3, Alastair Florence1*

1. CMAC, Strathclyde Institute of Pharmacy and Biomedical Sciences, University of

Strathclyde, Glasgow, UK

2. European Bioenergy Research Institute (EBRI), School of Engineering and Applied Science,

Aston University, Birmingham, UK

3. Chemical Engineering Department, American University of Sharjah, Sharjah, United Arab Emirates. * corresponding author: alastair.florence@strath.ac.uk

Abstract

More than 165 induction times of butyl paraben - ethanol solution in batch moving fluid

oscillation baffled crystallizer with various amplitudes (1 - 9 mm) and frequencies (1.0 – 9.0

Hz) have been determined to study the effect of COBR operating conditions on nucleation.

The induction time decreases with increasing amplitude and frequency at power density

below about 500 W/m3, however, a further increase of the frequency and amplitude leads

to an increase of the induction time. The interfacial energies and pre-exponential factors in

both homogeneous and heterogeneous nucleation are determined by Classical Nucleation

Theory at oscillatory frequency 2.0 Hz and amplitudes of 3 mm or 5 mm both with and

without net flow. To capture the shear rate conditions in oscillatory flow crystallisers, a

Large Eddy Simulation approach in Computational Fluid Dynamics framework is applied.

Under ideal conditions the shear rate distribution shows spatial and temporal periodicity

and radially symmetry. The spatial distributions of the shear rate indicate an increase of

average and maximum values of the shear rate with increasing amplitude and frequency. In

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continuous operation, net flow enhances the shear rate at most time points, promoting

nucleation. The mechanism of the shear rate influence on nucleation is discussed.

Introduction

Crystallization is a key process for the purification and isolation of active pharmaceutical

ingredients, many small organic and fine chemical compounds, and is most commonly

implemented as a batch operation. With the potential advantages [1, 2] of less down time,

more efficient use of energy and space, better heat transfer and mixing and lower risk of

process failures, continuous crystallization is of increasing interest for pharmaceutical

manufacturing [3]. Operating continuous multiphase processes for extended periods can

also pose challenges including instabilities introduced by variations in temperature and

encrustation. Continuous crystallization has been investigated in different platforms

recently [4], i.e. mixed-suspension mixed-product-removal (MSMPR) [5] [6] [7, 8] and plug

flow reactor [9] including continuous oscillatory flow crystallisers (COFC) [1, 10, 11]. Primary

nucleation is a complex poorly understood process and is a critical transformation in

controlling the crystal form, number and size of particles in a crystallization process. It is

therefore essential to understand the process conditions within a continuous reactor under

which primary nucleation can occur or can be avoided.

The influence of shear rate on crystallization has been reported in many systems. For

example the polymorphic form of carbamazepine can be altered depending on whether the

crystallized solution is quiescent or exposed to high shear [12]. Stirring rates effect the

metastable zone widths of L-glutamic acid [13] carbamazepine [12] and m-hydroxybenzoic

acid [14]. The effect of shear rate on nucleation is however complex considering the

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variational shear distribution of the solution in a crystallizer. Computational fluid dynamics

(CFD) models, particularly Large Eddy Simulation (LES) models [15, 16], are powerful tools

to study transient flow behaviour (e.g. the shear) in crystallizers [17]. Several computational

fluid dynamics simulations have been performed to understand velocity and shear

conditions in oscillatory flow reactors. Jian [18] analysed velocity fields (e.g. surface and

cycle averaged velocity and velocity ratios), and the velocity ratios of axial to radial velocity

components in the oscillatory baffled crystallizer are largely independent of the column

diameter. Manninen [19] employed two parameters including the axial dispersion

coefficient and the ratio of axial and transverse velocities in the CFD simulations and Ni [20]

showed 3-D numerical simulation of oscillatory flow (e.g. flow pattern and velocity vector) in

a baffled column and validated the results with experimental observations using digital

particle image velocimetry measurements. However the quantitative analysis and spatial

distributions of shear rate in an oscillatory flow crystallizer have not been reported.

Butyl paraben (butyl 4-hydroxybenzoate) shown in Figure 1, as with other parabens are

generally considered to be safe [21] and are widely used in pharmaceutical products and

cosmetics [22]. Butyl paraben has only one known polymorph [23] with a relative low

melting point 340.5 K [24]. In ethanol, butyl paraben has a very high solubility [25, 26] and

usually forms prismatic shape crystals [27, 28]. A relative low solid-liquid interfacial energy

[23, 28] was determined, from nucleation studies of butyl paraben in ethanol under equal

stirring rate conditions. In this work, induction time experiments have been performed in a

batch moving fluid oscillatory baffled crystallizer (MFOBC) and continuous oscillatory flow

crystallizer (COFC) to investigate the impact of supersaturation and shear conditions

(oscillatory amplitudes, frequencies and net flow rates) on the nucleation kinetics of this

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system. Classical Nucleation Theory has been applied to estimate and analyse the interfacial

energy and pre-exponential factor for butyl paraben - ethanol system and the effect of

shear rate quantified. The spatial shear rate distributions have been estimated using CFD

determined at different times during the period of oscillation. The shear rate distributions

are dependent on the sinusoidal velocity of the piston movement (a period of piston

velocity is 𝑇𝑉,) as well as the net flow rate. The average, middle and maximum shear rates

have been compared at time of 0.25 𝑇𝑉, 0.5 𝑇𝑉, 0.75 𝑇𝑉, and 𝑇𝑉 with induction time results

in both the MFOBC and COFC. Before nucleation the distributions of the clusters in the

solution influenced by different shear rates appears to result in the different nucleation

behaviours.

Figure 1. Molecular structure of butyl paraben

THEORY

Interfacial energy and pre-exponential factor

In Classical Nucleation Theory, the free energy change upon nucleation is the free energy

change of bringing molecules from the supersaturated solution into the cluster, subtracting

the interfacial energy of the cluster. Under the assumption of a spherical shape the

derivation leads to the basic equation, which defines the nucleation rate

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𝐽 = 𝐴𝑒𝑥𝑝 (−∆𝐺𝑐

𝑘𝑇) = 𝐴𝑒𝑥𝑝 (−

16𝜋𝜎3𝑣𝑚2

3𝑘3𝑇3(𝑙𝑛𝑆)2) (1)

where σ is the solid-liquid interfacial energy, 𝑣𝑚 is the volume of one solute molecule, 𝐴 is

the pre-exponential factor, k is the Boltzmann constant, T is nucleation temperature, S is the

supersaturation and ∆μ is the difference in chemical potential between the solute in

solution and in the crystalline bulk phase, which is usually estimated by actual and

equilibrium solute mole fraction in the solution ∆𝜇 = 𝑘𝑇𝑙𝑛𝑆 ≈ 𝑘𝑇𝑙𝑛𝑥

𝑥∗ . The nucleus is

defined as a crystalline particle of size sufficient for growth to be thermodynamically

favourable.

The induction time, tind, is the time period from the establishment of the supersaturated

state to the first observation of crystals in the solution and is assumed to contain three parts

28: a relaxation time or transient period 𝑡𝑟, the time required for formation of a stable

nucleus, 𝑡𝑛, and the time for a nucleus to grow to a detectable size, 𝑡𝑔. Usually it is assumed

that the relaxation time and the growth time are negligible and that the induction time is

inversely proportional to the nucleation rate in nucleation in the solution of volume V:

ln𝑡𝑖𝑛𝑑 = −ln𝐽𝐻𝑂𝑁V = −ln𝐴𝐻𝑂𝑁V +𝐵

𝑇3(ln𝑆)2 (2)

𝐵 =16𝜋𝜎𝐻𝑂𝑁

3 𝑣𝑚2

3𝑘3 (3)

Induction time experiments are usually evaluated by plotting ln 𝑡ind, versus 𝑇−3(𝑙𝑛𝑆)−2, for

determination of the interfacial free energy 𝜎 from the slope, B. The pre-exponential factor,

𝐴, is positively proportional to the attachment frequency and negatively proportional to the

energy barrier for diffusion [29] or desolvation [30].

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In some cases, there is a tendency for a decreasing slope, B, at decreasing supersaturation,

which could indicate a mechanism transition [31] from homogeneous nucleation (HON) to

heterogeneous nucleation (HEN),

𝐽𝐻𝐸𝑁 = 𝐴𝐻𝐸𝑁𝑒𝑥𝑝 (−16𝜋𝜎𝐻𝐸𝑁

3𝑣𝑚2

3𝑘3𝑇3(𝑙𝑛𝑆)2) (4)

and the activity coefficient of interfacial energy in heterogeneous nucleation can be

determined by comparing the interfacial energy in homogeneous nucleation with that in

heterogeneous nucleation and an activity coefficient of interfacial energy ( 0 < γ =

𝜎𝐻𝐸𝑁/𝜎𝐻𝑂𝑁 < 1) is employed in Equ. (1).

CFD model (Subgrid-scale model)

A large eddy simulation (LES) model is able to describe the hydrodynamics (e.g. velocity and

shear rate) of the liquid phase within both reactors. The model was solved using the

computational fluid dynamic (CFD) software ANSYS FLUENT Version 14 [32] .

The main model equations used to simulate the non-reactive flow in the OBC are:

Continuity equations:

𝜕(𝜌𝑙)

𝜕𝑡+ 𝛻(𝜌𝑙�� 𝑙) = 0 (5)

Momentum equations:

𝜕(𝜌𝑙�� 𝑙)

𝜕𝑡+ 𝛻(𝜌𝑙�� 𝑙�� 𝑙) = −𝛻𝑃 + 𝛻𝜏𝑙 + 𝜌𝑙�� (6)

where:

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𝜏𝑙 = (𝜆𝑙 −2

3𝜇𝑙) (𝛻 · �� 𝑙)𝐼 + 2𝜇𝑙𝑆𝑙 (7)

𝑆𝑙 =1

2(𝛻�� 𝑙 + (𝛻�� 𝑙)

𝑇) (8)

where 𝑙 represents liquid phase, 𝜌𝑙 is density of liquid, �� 𝑙 is velocity of liquid, 𝜇𝑙 is kinetic

viscosity of the liquid, 𝑆 is strain rate of liquid, 𝜏𝑙 is shear stress of liquid, �� is gravity of

liquid and 𝑃 is pressure of liquid. λl is bulk viscosity of liquid, I is unit vector, ∇u l is the

spatial derivative of velocity vector, (∇u l)T is the transpose of ∇u l.

The deviatoric part of the subgrid-scale stress tensor is modelled using a compressible form

of the Smagorinsky model:

𝜏𝑖𝑗 −1

3𝜏𝑘𝑘𝛿𝑖𝑗 = −2𝜇𝑡 (𝑆𝑖𝑗 −

1

3𝑆𝑘𝑘𝛿𝑖𝑗) (9)

The eddy viscosity is modelled as

μt = ρLs2|S| (10)

Where S is the rate-of-strain for the resolved scale which is calculated by |S| = √2sijsij and

the mixing length for subgrid scales Ls is defined as

𝐿𝑠 = 𝑚𝑖𝑛(𝜅𝑑, 𝐶𝑠𝛥) (11)

where μt is the subgrid-scale of the turbulent viscosity, 𝜏𝑘𝑘 is the isotropic part of the

subgrid-scale stresses, τij is the deviatoric part of the subgrid-scale stress tensor, Sij is rate-

of-strain tensor for sub-scale, ��𝑖𝑗 is the rate-of-strain tensor for the resolved scale, δij is

Kronecker delta function, κ is von Karman constant, d is the distance to the closest wall and

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Cs is the Smagorisky constant which value used in this study is 0.2. Δ is local grid scale,

computing according to the volume of the computational cell using Δ = V1/3.

Experimental work

Butyl paraben (BP) > 99.0 % mass purity was purchased from Aldrich and used without

further purification. Ethanol (E) of 99.8 % mass purity was purchased from Aldrich, and

distilled water was used.

Induction times for butyl paraben have been determined in the MFOBC and COFC at

different levels of supersaturation. Figure 2 a) and b) show the set-up of both platforms

used. Only one vertical jacketed glass column was used in the MFOBC connected to a PTFE

bellow (piston) via a temperature controlled glass U-bend, and these two sections were

separated by a nitrile rubber membrane. Oscillation was generated by a linear motor and

Xenus controller operating a PTFE Bellow. One jacketed glass column 700 mm long comprises

15 cells each with an internal diameter of 16.3 mm and 25.6 mm long. Neighbouring cells were

separated by anular baffles each with an orifice diameter of 7.6 mm and a height of 2.5 mm. All

of the glass was 1.8 mm thickness. An FBRM probe and a thermocouple were fixed in the middle

of the glass column via probe ports to observe the number of particles in the solution and the

temperature of the solution, respectively. For the COFC set up, two identical glass columns were

connected with a glass bend mounted horizontally. One end of the first column was connected

to the oscillator and the end of the second column was connected to a peristaltic pump

(Watson-Marlow 520S) to circulate the solution with a pump head rotation speed 15 and 20

rpm under temperature control back into the first column through the oscillation section (Figure

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2). Oscillation is controlled in the same as the MFOBC. Calibration of the oscillatory amplitudes

in the COFC was performed prior to carrying out the experiments.

Figure 2 Schematic equipment set up a) MFOBC and b) COFC combined with part of actual

glass column photo and part of CFD simulation result.

200 mL of butyl paraben in ethanol solution (2.490 g butyl paraben / g ethanol) with

saturation temperature of 24.2 °C was prepared in a 400 ml glass bottle. This was held in a

water bath at a constant temperature of 50.0 °C and stirred by a magnetic stirrer for 60 min

to insure that no undissolved paraben was present. The solution was quickly transferred

into the MFOBC which was sealed by a fexible membrane. Frequencies in the range 1.0 to

9.0 Hz and amplitudes from 1 to 9 mm were applied. In all the experiments, the butyl

paraben solution in the MFOBC was heated to 50.0 °C for 30 minutes before the solution

was cooled to 35.0 °C at a rate of 2.0 °C per minute. In order to prevent overcooling the

sample during rapid cooling to the target temperature, the final temperature reduction was

to between 19.0 - 22.0 °C at 1.0 °C per minute. Temperatures were controlled using

PharmaMV v4.0 control software which was also used to collect the data from the PAT

probes. Eight high resolution web cameras (Manhattan HD 760 Pro) were used at several

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different positions in the crystallizer with images captured every 1 s. The FBRM probe (G400

version), with a measurement range of 0.25−2000 μm, was operated with a measurement

duration of 2 s. Chord length distributions in the range 0−1000 μm were recorded with

average and total values recorded.

Table 1 Experimental conditions used to measure induction times at different

supersaturations in the MFOBC. Values show total numbers of experiments carried out with

the number of different supersaturations in parentheses.

𝒇 (Hz) 𝒙𝟎 (mm)

1 2 3 4 5 6 7 8 9

1.0 3(3) - 10(4) - - 14(5) - - 14(5)

2.0 - 2(2) 9(8) 12(5) - 3(2) - - -

3.0 - - 5(5) - 13(5) - - - -

4.0 12(6) 4(4) 9(5) - - - - 4(4) -

5.0 - 9(6) - 4(3) - - - 3(3) -

6.0 2(2) - - 4(3) - - - - -

7.0 - - - 4(4) - - 4(3) 3(3) -

8.0 - - - 5(4) 4(4) - - 5(5) -

9.0 - - 2(2) 2(2) - - - - 3(3)

𝑓: frequency, 𝑥0: amplitude, - : no experiment performed.

The nucleation time values, corresponding with the first observation of crystals from the

camera images, were measured. For FBRM nucleation was considered to have occurred at

the first increase in the counts in the range 0.25 - 1000 μm. The induction time was

determined as the time from the solution reaching constant temperature (supersaturation)

to the measured initial nucleation time combined the methods of camera images and FBRM

curves. After a nucleation event was measured, the solution was reheated to 15 oC above

the solubility and held for 30 minutes until all particles had redissolved. The cooling and

heating circulation were repeated multiple times varying the supersaturation and oscillatory

conditions. Experiments were carried out in a randomised order.

In total 168 nucleation experiments were performed in the MFOBC, shown in Table 1. Fewer

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replicates at higher frequencies were carried out due to rupturing of the oscillating

membrane after extended operation at these conditions. For the COFC experiments, 350mL

solutions were prepared and the experiments were performed in an identical manner to

those used in the MFOBC experiments. Totally 45 induction time measurements were

carried under the oscillatory and flow conditions shown in Table 2.

Table 2 Experimental conditions in MFOBC at frequency 2.0 Hz with amplitude 3 mm and in

COFC at frequency 2.0 Hz with amplitude 3 mm or 5 mm with different net flow rates.

Exp. 𝑓 (Hz) 𝑥0 (mm) Q (mL/min) V (mL) Experiments Sol.

I MFOBC 2.0 3 0 120 18 1*,2 II COFC 2.0 3 60.0 280 16 3,4 III COFC 2.0 5 0 280 16 4,5,6 IV COFC 2.0 5 100.0 280 13 4,7,8

𝑓: frequency, 𝑥0: amplitude, Q: pumping net flow rate, V: volume of solution, Concentration of Sol. (solution) 1-8: 2.490, 2.274, 2.050, 2.761, 2.541, 2.619, 2.569, 2.570 g butyl paraben / g ethanol, * same experiment of 2.0 Hz - 3 mm in Table 1.

Computation simulation procedure and boundary conditions

All CFD models were three-dimensional in order to capture possible chaotic behaviour. The

computational meshes were all hexahedral and structured. The computational cell size is

less than 1 mm on average, with denser mesh close to the walls. The model equations were

solved using the finite volume approach. First-order discretization schemes were used for

the solution of the convection terms in all governing equations. The relative error between

any two successive iterations was specified by using a convergence criterion of 1 × 10−3 for

all equations. The linearized equations for governing equations were solved using a block

algebraic multigrid method. In order to ensure easy convergence of the various partial

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differential equations (PDE) in the model, the Courant–Friedrichs–Lewy (CFL) condition for

three-dimensional PDE is followed:

𝐶 =𝑢𝑥∆𝑡

∆𝑥+

𝑢𝑦∆𝑡

∆𝑦 +

𝑢𝑧∆𝑡

∆𝑧 ≤ 𝐶𝑚𝑎𝑥 (12)

where 𝐶𝑚𝑎𝑥 is specified by the CFL condition to fall within the range of ~ 1 - 5 [32, 33]. In

this study, a time step of 0.001 seconds was found to satisfy this condition. The liquid

density ρ 1000 𝑘𝑔 ∙ 𝑚−3 and dynamic viscosity 𝜇𝑙 0.01613 𝑘𝑔 ∙ 𝑚−1𝑠−1, determined at 25°C

with concentration of 2.5 g butyl paraben / g ethanol solution, was used in the simulation.

The OFC consist of a repeating series of cells as shown in Figure 2. The sinusoidal oscillatory

movement is provided by a piston at one end of the reactor. The displacement, zp and the

velocity,uz, of the piston are given by

zp = −𝑥0cos (2πft) (13)

uz = 2πf𝑥0sin (2πft) (14)

Results

Induction time in MFOBC as a function of oscillation

Figure 3 shows images of the butyl paraben solution inside the MFOBC (top row) and COFC

(bottom row) and the evolution of their appearance from before (a) and after nucleation (b

and c). The initial appearance of crystals with the subsequent increase in turbidity indicated

the nucleation occurring. Following nucleation in both reactors a rapid increase in turbidity

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occurs due to secondary nucleation. Seeding experiments in a system of butyl paraben in

ethanol show the solution becomes totally opaque in under 10 seconds after the addition of

fine crystalline seeds particles under moderate shear rate. Compared with the length of

induction (minutes or longer), the assumption is that influence of secondary nucleation has

a limited influence on the induction time measurements in this work. In the COFC, the

circulating solution through the pump is less than 5% of the volume of solution in the

crystallizer, and the shear rate at the pump head speeds used is relatively low. Imaging does

not show any evidence of primary nucleation in this section of the reactor. Therefore, the

influence of the solution inside the recirculation section on the overall nucleation kinetics

can be neglected. The FBRM data and more imaging data used to determine the induction

times are provided as supporting information.

a) b) c)

Figure 3 The progress of nucleation in the MFOBC (top row) and COFC (bottom row) after

holding at a fixed supersaturation. Column a): both solutions prior to nucleation with the

high contrast writing clearly visible through the solution. b): < 5 seconds after the observed

onset of nucleation with a significant increase in turbidity and partial obscuration of the

2 mm 2 mm 2 mm

5 mm 5 mm 5 mm

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writing. c): < 25 seconds after nucleation with the rapid appearance of crystals to form a

highly turbid suspension.

The natural logarithm of the induction times measured from each experiment at

frequencies between 1.0 and 9.0 Hz are plotted vs RTlnS to show the range of nucleation

times observed across the different frequencies and amplitudes investigated in this study

(Figure 4). At equal oscillation conditions the induction time increases with decreasing

driving force (RTlnS) as expected. Whilst most of the lines show similar gradients, some of

the lines show an obvious deviation due to the limited number of experiments performed at

those individual oscillatory conditions. There is an obvious effect on the induction time at

equal driving force from both frequency and amplitude. At each frequency tested from 1.0

Hz to 6.0 Hz (top two rows in Figure 4) the induction times at comparable values of RTlnS

tend to decrease with increasing amplitude. At the lowest frequency, 1.0 Hz, the reduction

in induction time as amplitude increases from 1 to 9 mm is greater than that observed at

any other frequency. Notably, above 7.0 Hz the induction times tends to increase with

increasing amplitude for comparable RTlnS values (bottom row in Figure 4).

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Figure 4 Induction time (lntind) vs. driving force (RTlnS) of nucleation in the MFOBC under

different amplitudes with each frequency, respectively. Legend: Frequency (Hz) – amplitude

(mm). Dashed lines are fitted lines for the linear correlation of the respective experimental

data.

Induction times under different frequencies for each value of amplitude tested are shown in

the supporting information. At each oscillatory amplitude from 1 mm to 6 mm the induction

times tend to decrease with increase of the frequency under comparable RTlnS values, and

at the lowest amplitude, 1 mm, the decreased induction time with increase of the frequency

is higher than at other amplitudes. However, at 8 mm the induction times tend to increase

with increasing frequency from 5.0 Hz to 8.0 Hz under comparable equal driving force. A

plot of all induction time results vs. RTlnS (shown in the supporting information) shows a

much stronger influence of oscillation conditions at lower amplitudes and frequencies than

in the high region, which is consistent with Figure 4. This tendency is in agreement with the

3

6

9

1.0-1

1.0-3

1.0-6

1.0-9

2.0-2

2.0-3

2.0-4

2.0-6

3.0-3

3.0-5

3

6

9

4.0-1

4.0-2

4.0-3

4.0-8

5.0-2

5.0-4

5.0-8

6.0-1

6.0-4

100 200 300

3

6

9

7.0-4

7.0-7

7.0-8

100 200 300

8.0-4

8.0-5

8.0-8

100 200 300

9.0-3

9.0-4

9.0-9

RTlnS (J/mol)

lnt in

d

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simulation results [34] that nucleation behaviour is more sensitive to the shear rate at lower

shear rate regime than at higher shear rate regime.

Figure 5 Estimated induction time at driving force (RTlnS) 200 J/mol of butyl paraben in

ethanol with various frequencies and amplitudes in the MFOBC.

The induction times at RTlnS 200 J/mol were estimated from the linear correlation lines in

Figure 4 at each set of oscillatory conditions, and the relation between the extrapolated

induction time with amplitude and frequency is shown as a response surface in Figure 5. The

longest induction time (region A, red, Figure 5) is almost three orders of magnitude greater

than the shortest induction time. Increasing the frequency and amplitude above 1.0 Hz / 1

mm shows a significant decrease in the induction time in region A. Similar values of

induction time are found across a range of oscillatory conditions (region C, green, Figure 5).

The minimum values for induction time of all conditions are shown between 5-7 Hz and 5-7

mm (point B, purple, Figure 5). A similar tendency of the nucleation behaviour was also

A

B

C

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reported for butyl paraben in ethanol solution in a stirred glass vessel between 100 – 800

rpm, which displayed a minimum induction time for stirring rates around 200 rpm [35].

The power density, 𝑃

𝑉, Reynolds number, 𝑅𝑒0

and Strouhal number 𝑆𝑡 of an oscillation

crystallizer are calculated [10] as:

𝑃

𝑉=

2𝜌𝑙𝑁𝑏

3𝜋𝐶𝐷2 (

𝐴12−𝐴2

2

𝐴22 )(2𝜋𝑓𝑥0)

3 (15)

𝑅𝑒0=

2𝜋𝑓𝑥0𝜌𝑙𝐷

𝜇𝑙(𝐴1

2−𝐴22

𝐴22 )(2𝜋𝑓𝑥0)

3 (16)

𝑆𝑡 =𝐷

4𝜋𝑥0 (17)

where 𝑁𝑏 is the number of baffles per unit length, CD is the coefficient of discharge of the

baffles (~ 0.7), 𝐴1 and 𝐴2 is cross sectional area of the tube and of the baffle orifice,

respectively. D is the internal diameter of the crystallizer. From equations (15), (16) and (17),

power density, 𝑅𝑒0 and 𝑆𝑡 can be estimated in the range of frequency 1 Hz ≤ 𝑓 ≤ 9 Hz and

amplitude 1 mm ≤ 𝑥0 ≤ 9 mm . Figure 6 shows that induction time decreases with

increasing power density and tind ∝ (P

V)−0.59 when power density is below 500 W/m3

(𝑅𝑒0~200). A similar relationship has been reported for the nucleation kinetics of butyl

paraben in ethanol solution in small stirred glass vessels [35]. As power density is increased

above 1000 𝑊/m3 (𝑅𝑒0~300), the induction time starts to increase. It has been reported

that at 𝑅𝑒0 values above 300 a more turbulent, well mixed flow scheme develops within

oscillatory baffled reactors [36]. In this high shear range, the induction time shows a positive

correlation with power density with 𝑡𝑖𝑛𝑑 ∝ (P

V)0.46 (Figure 6).

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Figure 6 Estimated induction time at RTlnS = 200 J/mol versus power density in the MFOBC.

Dashed lines show best fit to the respective data, black < ln(P/V) = 6.2 < red.

Strouhal number is the ratio of column diameter to stroke length, and in this work, the

Strouhal numbers are in the range 1.3 to 0.2 indicating a collective oscillating movement of

the fluid "plug" [37] regardless of the size of the Reynolds number. For the experimental

conditions with equivalent power densities, the induction time tends to decrease with

increasing amplitude (shown in supporting information), i.e. decreasing Strouhal number.

Exceptions to this trend are 6.0 Hz - 1 mm, 4.0 Hz - 8 mm and 8.0 Hz – 4 mm. Further data

under these conditions would be required to assess the statistical significance of these

particular results.

Simulated hydrodynamics in MFOBC and COFC

A LES model based CFD simulation was implemented to simulate the hydrodynamics of the

liquid phase in the COFC geometry (horizontal arrangement) at a range of flow rates, Q = 0,

60 and 100 mL/min. 0 mL/min is equivalent to the conditions within a batch operated

MFOBC (i.e. frequency and amplitude with no net flow) whilst the values of Q > 0 are

-3 0 3 6 9

3

6

9

lnt in

d

ln(P/V)

19

equivalent to a continuously operated COFC. The simulated results showed the reactor cell

has periodic symmetry of the flow pattern after just few seconds of fluctuation, and

accordingly 3 cells are used to represent 15 cells in the experimental setup (Figure 7). The

simulation results indicate that the flow within each oscillatory baffled crystallizer is subject

to both temporal and spatial periodicity. The spatial periodicity is shown in Figure 7a and

Figure 7c, which are snapshots of the velocity and shear rate spatial distribution,

respectively. The velocity and shear rates increase or decrease along the radial direction.

The temporal periodicities are shown in Figure 7b and Figure 7d and highlight the evolution

of axial velocity and shear rate at point d (in the high shear rate region), respectively. Figure

7 shows an example of the simulation results at 0.5n s (n is an integral number) when the

piston velocity is zero and located at the negative maximum distance (amplitude distance,

mm). Due to the periodic movement of the piston, the flow of liquid is accelerated as it

passes the baffles. Therefore, a maximum axial velocity appears around the baffle

connecting two neighbouring cells. The shear rate is directly related to the gradient or

derivative of velocity, and the maximum shear rate appears where the deformation of liquid

is large, particularly near the baffles. The simulated eddies appear before and after the

solution passes through the baffle orifice, indicating intensive turbulence in this region,

which is consistent with the experimental observation. The influence on the shear rate and

velocity distributions of moving from a horizontal to a vertical orientation were also

calculated under identical oscillation conditions. The simulation results show the influence

of gravity is negligible.

20

Figure 7 The 3D CFD simulated hydrodynamic behaviours for a liquid phase at 2.0 Hz - 3mm -

60ml/min. A. velocity vector at 0.5n s, B. temporal axial velocity at monitor point d in the

centre of the baffle orifice, C. shear rate vector at point d at 0.5n s and D. temporal shear

rate at point d.

Simulations I, II, III and IV were at the same oscillation conditions as Experiment I, II, III and

IV (Table 2), respectively, but with horizontal COFC employed in all the simulations. Net flow

causes an increase in the velocity curve at point d (Figure 7) at both 2.0 Hz – 3mm and 2.0

Hz – 5 mm cases (Figure 8). Without net flow applied the velocity at point d is positive for 50%

of the oscillatory period of the piston movement and rises to 75% of the period of the piston

oscillation with a net flow applied. When the piston movement is aligned with the direction

of net flow the solution velocity is enhanced, and accordingly shear rate increases. When

the piston movement is in the opposite direction to the net flow the velocity is reduced.

9.6×10-2

0 (m/s)

Velocity a b c

A)

B)

d

a b c

d

C)

D)

30

0 (1/s)

Shear rate

Flow times (s)

She

ar r

ate

(1/s

)

Flow times (s)

Axi

al v

elo

city

(m/s

)

At point d

At point d

0

0.1

0 4 8

0 4 8

10

2

0

21

Figure 8 Simulated velocity of flow at point d in same period of piston velocity

The accumulated spatial distributions of the shear rate at each time point 0.25 𝑡𝑉, 0.5 𝑡𝑉,

0.75 𝑡𝑉 and 𝑡𝑉 within a period of the piston velocity are shown in Figure 9. The overall shear

rate at each time point is generally higher at 5 mm amplitude than at 3 mm. This indicates a

stronger influence of a 2 mm increase in oscillation amplitude compared with a 60 mL/min

increase of net flow on these two distributions. However at 0.5 𝑡𝑉 the shear rate

distribution at 3 mm amplitude and 60 mL/min net flow is very slightly higher than at 5 mm

amplitude and 0 mL/min. From the shear rate distribution, the shear rate with the largest

volume fraction from simulation I is 1.3 𝑠−1 at 0.5 𝑡𝑉 with a volume fraction about 19.5 %,

for simulation II it is 1.3 𝑠−1 at 𝑡𝑉 with volume fraction about 13.3 %, simulation III is 12.5

𝑠−1 at 𝑡𝑉 with a volume fraction of 14.9 %, and in simulation IV the value is 12.5 𝑠−1 at 𝑡𝑉

with a volume fraction about 24.8 %. For simulations I, II, III and IV, the median value of the

shear rate distribution is in the range of 1.8 – 4.3 𝑠−1, 2.5 - 5.5 𝑠−1, 3.5 - 7.0 𝑠−1, and 4.5 -

11.2 𝑠−1 respectively. The maximum shear rate for the highest 10% volume fraction is in the

range 12.8 - 52.5 𝑠−1, 17.5 - 87.5 𝑠−1, 22.5 – 97.5 𝑠−1 and 32.5 - 150.0 𝑠−1. They are all in

order simulation I < II < III < IV. The increase in standard deviation of shear rate at the four

22

time point 0.25 𝑡𝑉 , 0.5 𝑡𝑉 , 0.75 𝑡𝑉 and 𝑡𝑉 follows the same order, indicating a larger

fluctuation of spatial and temporal shear rate with increasing shear rate (higher amplitude

or higher net flow; Table 4).

Figure 9 Accumulated volume distributions of the shear rate at different time points of the

oscillatory flow, specifically 0.25 tV, 0.5 tV, 0.75 tV and tV of a period of piston velocity. Each

curve shows the accumulated volume distribution for different operation conditions

denoted as [frequency - amplitude – net flow rate]: (black) 2.0 Hz – 3 mm – 0 mL/min (red)

2.0 Hz - 3 mm - 60 ml/min (blue) 2.0 Hz - 5 mm - 0 ml/min (green) 2.0 Hz – 5 mm – 100

ml/min

At 0 mL/min the accumulated spatial shear rate distributions are almost overlapping at 0.25

𝑡𝑉 and 0.75 𝑡𝑉 (supporting information), where the piston velocity is at a maximum, and are

almost overlapping at 0.5 𝑡𝑉 and 𝑡𝑉, where the velocity of the piston at the inlet is zero.

When net flow is applied, the shear rate distributions become more complicated.

0

20

40

60

80

100

Accu

mula

tive d

istr

ibu

tion

f(Hz)-X0(mm)-Q(mL/min)

2.0-3-0

2.0-3-60

2.0-5-0

2.0-5-100

f(Hz)-X0(mm)-Q(mL/min)

2.0-3-0

2.0-3-60

2.0-5-0

2.0-5-100

0.1 1 10 100

0

20

40

60

80

100

Accu

mula

tive d

istr

ibu

tion

Shear rate (s-1)

f(Hz)-X0(mm)-Q(mL/min)

2.0-3-0

2.0-3-60

2.0-5-0

2.0-5-100

0.1 1 10 100

Shear rate (s-1)

f(Hz)-X0(mm)-Q(mL/min)

2.0-3-0

2.0-3-60

2.0-5-0

2.0-5-100

0.0 0.5 1.0

-1

0

1 0.25tV

U/U

0

tV

0.0 0.5 1.0

-1

0

1

0.5tVU

/U0

tV

0.0 0.5 1.0

-1

0

1

0.75tV

U/U

0

tV

0.0 0.5 1.0

-1

0

1

tV

tVU

/U0

23

Comparing simulations II and III, a similar tendency is captured: a) the shear rate at 0.25 𝑡𝑉

is higher than 0.5 𝑡𝑉 which are both higher than the shear rate at other time points, b) the

shear rate at 𝑡𝑉 is higher than 0.75 𝑡𝑉 in the high shear region (above 15 s-1) and the shear

rate at 𝑡𝑉 is lower than 0.75 𝑡𝑉 in the lower shear region (supporting information). This is

consistent with the average shear rates shown in Table 3. In simulations I and III the highest

and lowest average shear rate alternately appears every 0.25 𝑡𝑉. Whereas in simulations II

and IV the highest average shear rate and the lowest average shear appears at 0.25 𝑡𝑉 and

𝑡𝑉, respectively. The difference of shear rate among simulation I – IV is smallest at time

point 𝑡𝑉, which is highest at time point 0.25 𝑡𝑉 shown in Table 3.

Table 3 Average and max shear rate in simulation I, II, III and IV with standard deviations

Simulation

Conditions f(Hz)-X0(mm)-Q(mL/min)

Average shear rate (s-1) at time point

10% maximum shear rate of the volume

distribution (s-1) 0.25

𝑡𝑉 0.5 𝑡𝑉

0.75

𝑡𝑉 𝑡𝑉

Standard deviation for each simulation

I 2.0-3-0 5.11 2.34 5.11 2.34 1.39 12.75-52.50

II 2.0-3-60 7.54 4.87 3.55 3.21 1.70 17.50-87.50 III 2.0-5-0 8.98 4.62 8.98 4.62 2.12 22.50-97.50

IV 2.0-5-100 14.60 10.25 7.33 5.46 3.45 32.50-150.00 Standard deviation

at each time point 3.49 2.90 2.07 1.21

Nucleation parameters in MFOBC and COFC

Figure 10 shows the experimental results, induction times, lntind, of solutions 1 – 8 with

different concentrations in the MFOBC and COFC platforms (Table 2) versus 𝑇−3ln𝑆−2. The

linear regression solid and dashed lines in Figure 10 for determining the interfacial energy

and pre-exponential factor in the homogeneous nucleation and in the heterogeneous

nucleation are nearly parallel to each other, respectively. In Figure 10, the induction time

24

decreased (Exp. I > Exp. II > Exp. III > Exp. IV) under comparable supersaturation in both

homogeneous nucleation and heterogeneous nucleation due to the influence of the

oscillatory conditions and the net flow rates by changing the shearing condition, as well as

the influence of the solution volume which is consistent with that metastable zone widths

decrease with increasing of solution volumes [34, 38, 39]. Because of the stochastic nature

of the nucleation processing, the average values of 𝑅2 for all the linear correlation are about

0.83, where the 𝑅2 values are overall lower in the heterogeneous nucleation (dashed lines)

than in the homogeneous nucleation (solid lines).

In the homogeneous nucleation region, the solid-liquid interfacial energies, σHON, of butyl

paraben - ethanol in the MFOBC at 2.0 Hz – 3 mm is 1.137 mJ ∙ m−2, determined by Equ. (2),

and in the COFC the interfacial energies are 1.134, 1.094 and 1.051 mJ ∙ m−2 for 2.0 Hz – 3

mm at 60 mL/min and 2.0 Hz – 5 mm at 0 mL/min and 100 mL/min, respectively. The

interfacial energies in the homogeneous nucleation in these experiments are in good

agreement, with a standard deviation of only 0.0350 mJ ∙ m−2, and the values are also

consistent with the values, 1.14 - 1.16 mJ ∙ m−2, measured in other platforms [23, 35]. The

interfacial energy, σHEN, in the heterogeneous nucleation are determined by Equ. (4) shown

in Table 4, which are in the range of about 0.50 mJ ∙ m−2 to 0.55 mJ ∙ m−2 with the activity

coefficient factor about 0.46 – 0.50, and the standard deviations also indicate the

consistence between the interfacial energies in the heterogeneous nucleation in these

experiments. From inspection of the intersections of the fitted lines in Figure 10, at higher

shear rates the nucleation mechanisms tends to shift to a heterogeneous regime at a slightly

lower driving force, and this tendency is also in agreement with butyl paraben in ethanol as

25

a function of increasing stirring rate [35]. These indicate under higher shear rate

heterogeneous nucleation become a little more unfavourable.

Figure 10 ln induction time (lntind) vs. 𝑇−3(ln𝑆)−2 in the MFOBC and COFC. Dashed and solid

lines are best fitting linear lines for the respective data.

26

Table 4 Interfacial energy and pre-exponential factor of butyl paraben-ethanol in

homogeneous and heterogeneous nucleation

Exp. 𝜎𝐻𝑂𝑁 (mJ ∙m−2)

𝐴𝐻𝑂𝑁𝑉 (s−1)

𝜎𝐻𝐸𝑁 (mJ ∙m−2)

𝐴𝐻𝐸𝑁𝑉 (s−1)

𝛾 |ṙ|

(s−1) |𝑢|

(m ∙ s−1)

I 1.137 0.038 0.551 0.0017 0.48 3.73 0.037

II 1.134 0.056 0.515 0.0024 0.46 4.79 0.043

III 1.094 0.110 0.548 0.0049 0.50 6.80 0.059

IV 1.051 0.116 0.503 0.0067 0.48 9.41 0.068

Standard deviation

0.035

0.021

0.02

γ =σHEN

σHON⁄ , |ṙ|: average shear rate at four time points (0.25 tV, 0.5 tV, 0.75 tV and tV)

at each experiment. |𝑢| : average absolute velocity during a period of piston velocity

Figure 11 Relation between pre-exponential factors with average shear rates with guiding

dashed lines. Experiment I to IV (Table 2) from left to right.

The calculated pre-exponential factors for homogeneous nucleation of butyl paraben are

relatively low, which have similar orders of magnitude as those reported in previous

experiments [34]. The pre-exponential factor, 𝐴𝑉 , in both the homogeneous and

heterogeneous nucleation regimes increases in the order Exp. I < Exp. II < Exp. III < Exp. IV

27

with 𝐴𝐻𝐸𝑁𝑉 nearly one or two orders of magnitude lower than the value of 𝐴𝐻𝑂𝑁𝑉 across

all conditions. This may be due to the lower number of potential nucleation sites available in

heterogeneous nucleation (foreign particles) compared with homogeneous nucleation

(molecules/molecular clusters) [29]. The pre-exponential factors, 𝐴𝑉 , increase with

averaged shear rates (from the CFD simulation results) in homogeneous and heterogeneous

nucleation, respectively (Figure 11). This is due to enhancing the attachment frequency and

probably decreasing the energy barrier for diffusion or desolvation. The shear rate has a

stronger influence on the pre-exponential factors in heterogeneous nucleation (𝐴𝐻𝐸𝑁𝑉 in

Exp. IV is about 4 times than in Exp. I) than in homogeneous nucleation (𝐴𝐻𝑂𝑁𝑉 in Exp. IV is

about 3 times than in Exp. I).

Discussion

The interfacial energy values in homogeneous nucleation determined in this work are

consistent with each other as well as with values from other studies [23, 35]. As expected,

the critical radius of the cluster decreases with increasing driving force / supersaturation at

each of the four mixing conditions tested. The variation in critical nucleus size under

different mixing conditions also tends to reduce with increased supersaturation. This may

be as a result of reduced sensitivity of smaller critical nuclei to changes in average shear

under experimental conditions in Exp. I - IV, compared with larger clusters comprising more

molecules obtained at lower supersaturations. The same tendency has been observed in

butyl paraben - ethanol solution in a Taylor-Couette flow system [35].

28

The influence of shear rate on metastable zone width and induction time over limited

ranges [41, 42] can be explained in several ways including enhanced mass transfer of

molecules to the cluster, inducing molecular ordering [43] or by promoting secondary

nucleation following an initial primary nucleation event [42] [44]. The results from this work

(Figures 5 and 6) and from other systems including L-glutamic Acid [45], 𝐻𝑁3𝐻2𝑃𝑂4, 𝑀𝑔𝑆𝑂4,

𝑁𝑎𝑁𝑂4 [46, 47] and H2O [48] emphasise that the interaction between nucleation rate and

shear rates over extended ranges can vary quite significantly. This is perhaps to be expected

in light of the dependency of cluster growth, coalescence and breakup that can combine to

lead to an increase or decrease in nucleation rate as shear rate increases [47], [48].

Figure 12 Schematic influence of shear rate on the induction time by effecting the

distribution of the clusters

29

CNT assumes a distribution of prenucleation molecular clusters spanning single molecules, n

= 1, to critical nuclei, n*. During nucleation the chemical potential drives the growth of

clusters through coalescence and addition to form a stable nucleus (n*) [49]. Molecular

dynamics simulations show that average shear rate can change the size distribution of

molecular clusters in solution and both enhance or suppress the nucleation rate depending

on the value [50]. Many experimental studies and simulations reported that an increase in

the shear rate leads to the decrease in size distribution of particles in the solution [51-54]. In

terms of explaining the observed nucleation kinetics for butyl paraben, at low shear rates

(Region I, Figure 12) molecular clusters grow relatively slowly with the cluster size

distribution taking comparatively longer for n* clusters to emerge compared with higher

shear rates. In Region II (Figure 12) the highest nucleation rates are observed, with the rate

of change of the cluster size distribution at a maximum. Further increases in shear rate take

the system away from the minimum in nucleation time (Region III). This suppression in

growth rate of the clusters may be due to the fact that larger clusters approaching n* are

now disrupted. Hence shear enhances nucleation rate overall, with all values greater than

no shear, but that beyond the optimum value where growth of clusters towards n* is at a

maximum, shear rate can remove some of the larger clusters, biasing the distribution

towards lower values of n.

The CFD simulation indicates that spatial symmetry of the shear rate distributions in

individual cells of the OFCs is retained independent of the length of reactor used (i.e. the

number of glass straights used). For each glass column the shear rate distributions are equal

and depend only on the oscillatory conditions and solution properties. Theoretically

therefore the shear rate distribution is readily predicted for different lengths of reactor

30

based on the conditions in a single glass column. The consistency of the simulated increase

in shear rate between the MFOBC and COFC is supported by the reduced experimental

induction time results at equivalent driving force. In a stirred tank however, the shear rate

distributions are more complex upon scale up. Therefore, the OFC platform shows relative

simplicity in scaling up from small scale nucleation experiments in a MFOBC for example to

deliver consistent performance in a larger length COFC [18, 55].

Conclusions

Induction time measurements and simulated shear rate distributions are reported for the

first time in an OFC. The nucleation kinetics for butyl paraben in ethanol show a strong

dependency on the process conditions, specifically oscillation amplitude, frequency and net

flow. The CFD simulations provide an explanation for the differences in experimental

induction times observed, highlighting the importance of changes in flow on the resultant

shear rate distribution. As expected, the highest shear rate appears in close proximity with

the baffles in both OFC configurations tested. Critically the simulations inform the

interpretation of variation in induction times as a function of shear rate. This attributes to

the impact of shear rate on the dynamics of prenucleation cluster growth in the context of

CNT. For equivalent power densities, increasing amplitude or reducing frequency leads to

shorter induction times. The simulation and experimental results show consistency between

batch MFOBC and COFC and stronger influence of increasing the amplitude than increasing

the net flow. The addition of net flow in COFC is shown to decrease the induction time by

increasing the pre-exponential factor in both homogeneous and heterogeneous nucleation

regimes due to an increase in the average shear rate. Nucleation behaviours under various

shear conditions show that the observed nucleation rates go through a maximum as shear

31

rate increases. For butyl paraben, which shows a relatively narrow metastable zone width in

ethanol, the ability to control and optimise nucleation rates in oscillatory flow opens the

potential to exploit OFCs to control primary nucleation for the design of continuous

crystallisation processes. Moreover the results provide a rational basis for the conversion of

batch OFC crystallisation to a continuous OFC through the detailed understanding of the

influence of net flow on nucleation.

Notations

𝐴 Pre-exponential factor [s-1•m-3]

𝐴𝐻𝐸𝑁 Pre-exponential factor in heterogeneous nucleation [s-1•m-3]

𝐴𝐻𝑂𝑁 Pre-exponential factor in homogeneous nucleation [s-1•m-3]

B Slope of ln𝑡𝑖𝑛𝑑 equation [K3]

𝐶𝑠 Smagorinsky constant, 0.2

𝐷 Distance to the closest wall [m]

𝑓 Frequency [Hz] 𝐽 Nucleation rate [number • m-3•s-1]

𝐽𝐻𝐸𝑁 Nucleation rate in heterogeneous nucleation [number • m-3•s-1]

𝐽𝐻𝑂𝑁 Nucleation rate in homogeneous nucleation [number • m-3•s-1]

𝑘 Frequency [Hz]

𝛫

Von Karman constant M Molecular weight [g•mol-1]

NA Avogadro constant, 6.022×1023 ［mol-1]

𝑃 Pressure of liquid [kg•m-1•s-2] Q Net flow rate [mL•min-1] 𝑟𝑐 Critical nuclei radius ［nm]

R Gas constant, 8.3145 [J•mol-1•K-1]

Supersaturation

𝑆𝑖𝑗 Rate-of-strain tensor for sub-scale. [kg•m-1•s-2]

𝑡𝑖𝑛𝑑 Induction time of nucleation [s]

𝑡 Piston movement time [s] T Temperature [K]

𝑇𝑉 A period of piston velocity [s]

𝑉 Solution volume [m3]

𝑣𝑚 Molecular volume [m3]

𝑥 Actual solute molar fraction solubility [mol•mol-1 total]

𝑥0 Amplitude [mm] 𝑥∗ Equilibrium solute molar fraction solubility [mol•mol-1 total]

𝑧𝑝 Velocity of piston [m•s-1]

𝜆𝑙 Bulk viscosity of liquid [pa] 𝛾 Activity coefficient of interfacial energy 𝜎 Solid-liquid interfacial energy [mJ•m-2]

𝜎𝐻𝑂𝑁 Interfacial energy in homogeneous nucleation [mJ•m-2]

𝜎𝐻𝐸𝑁

Interfacial energy in heterogeneous nucleation [mJ•m-2]

𝜋 3.1416

S

32

𝜇𝑡 Subgrid-scale turbulent viscosity [kg•m-1•s-1]

𝑢𝑧 Position of the piston

𝜌𝑙 Density of liquid [kg•m-3]

𝜏𝑘𝑘 Isotropic part of the subgrid-scale stresses [kg•m-1•s-2]

𝛿𝑖𝑗 Kronecker delta function 𝜏𝑖𝑗 Deviatoric part of the subgrid-scale stress tensor [kg•m-1•s-2]

𝜇𝑙 Kinetic viscosity of the liquid [m2•s-1]

𝑆 Rate-of-strain for the resolved scale [s-1] �� 𝑙 Velocity vector of liquid [m•s-1]

𝑆 Strain rate of liquid [s-1]

𝜏𝑙 Shear stress of liquid [kg•m-1•s-2]

�� Gravity of liquid [m•s-2] ��𝑖𝑗 Rate-of-strain tensor for the resolved scale [kg•m-1•s-2]

𝐼 Unit vector

��𝑖𝑗 Rate-of-strain tensor for the resolved scale [s-1]

𝛥 local grid scale [m] ∆𝐺𝑐 Critical free energy of nucleation [kJ•mol-1] ∆𝜇 Driving force of nucleation [kJ•mol-1] 𝛻�� 𝑙 Spatial derivative of velocity vector [s-1] (𝛻�� 𝑙)

𝑇 Transpose of ∇u l [s-1]

Acknowledgments

We acknowledge the EPSRC Centre for Innovative Manufacturing in Continuous Manufacturing and

Crystallisation for funding, grant ref: EP/I033459/1. Dr Xi Yu gratefully acknowledges the research

fellowship from Leverhulme Trust and research grant (RPG-410).

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For Table of Contents Use Only

The effect of oscillatory flow on nucleation kinetics of butyl paraben

Huaiyu Yang, Xi Yu, Vishal Raval, Yassir Makkawi, Alastair Florence

Nucleation behaviours of butyl paraben - ethanol solution in oscillatory flow crystallizers

(OFC) with various amplitudes, frequencies and net pumping flow

Spatial distributions of the shear rate by CFD simulation inside OFCs

Maximum nucleation rate with increasing shear rate, due to the influence on the cluster

distribution before nucleation

Providing a rational basis for the conversion of batch OFC crystallisation to a continuous OFC